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A partial differential equation has usually a regular solution at the initialtime if the initial condition is smooth in space, fulfills the governingequations and is compatible with the boundary condition. In the case ofNavier-Stokes equation, the initial velocity field must also bedivergence--free. It is common belief that the initial condition is compatiblewith the boundary condition if the initial condition fulfills the boundarycondition but this is not sufficient. Such a field does not necessarily fulfillthe full compatibility condition of the Navier-Stokes equation. If thecondition is violated, the solution is not regular at the initial time. Thisissue has been known for a while but not in the full breadth of the fluiddynamics community. In this paper, a practical calculation method is presentedfor checking the compatibility condition. Furthermore, a smooth initialcondition is presented in a periodic channel flow that violates the conditionand has no spatially smooth solution at initial time. The calculations werewere performed in an analytical framework. The results for a channelconfiguration show that in the absence of wall--normal velocity the conditionis always fulfilled and the problem has a regular solution. If the wall--normalvelocity component is non--zero, the condition is usually not fulfilled butwith optimization methods counter-examples can be generated. The generationprocedure of such a field is useful to provide correct initial conditions forsensitive time-dependent numerical simulations. The presented methods canprovide insight about the applicability of the chosen initial conditions.

Analytic investigation of the compatibility condition and the initial  evolution of a smooth velocity field for the Navier-Stokes equation in a  channel configuration